# Limits without log

1. Mar 19, 2010

### wimma

1. The problem statement, all variables and given/known data
Find the limit as n--> infinity of (1-2/n)^n

2. Relevant equations

We know (1+1/x)^x --> e as n--> infinity

3. The attempt at a solution

I worked it out as e^(-2) using log but I can't get it out using the fundamental limit above. I know it's the square of (1-1/x)^x (where we let x=n/2), just I don't know how to show that (1-1/x)^x --> 1/e. If you could let x |--> -x somehow I'd get the desired result using the limit laws but I'm not sure that's allowed.

2. Mar 19, 2010

### Staff: Mentor

Let n = -2x. This makes your limit
$$\lim_{-2x \to \infty} (1 + \frac{1}{x})^{-2x}$$

With a bit of adjustment you can use the limit you know.

3. Mar 19, 2010

### wimma

but won't the parameter go to -infinity so we can't equate (1+1/x)^x to e?

4. Mar 19, 2010

### Staff: Mentor

As it turns out,
$$\lim_{x \to -\infty} (1 + \frac{1}{x})^x~=~e$$

Can you use this fact?